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Mathematics of Computation

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Closed-form summation of some trigonometric series

Authors: Djurdje Cvijović and Jacek Klinowski
Journal: Math. Comp. 64 (1995), 205-210
MSC: Primary 65B10; Secondary 33E20, 65D20
MathSciNet review: 1270616
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Abstract: The problem of numerical evaluation of the classical trigonometric series

$\displaystyle {S_\nu }(\alpha ) = \sum\limits_{k = 0}^\infty {\frac{{\sin (2k +...\limits_{k = 0}^\infty {\frac{{\cos (2k + 1)\alpha }}{{{{(2k + 1)}^\nu }}},} $

where $ \nu > 1$ in the case of $ {S_{2n}}(\alpha )$ and $ {C_{2n + 1}}(\alpha )$ with $ n = 1,2,3, \ldots $ has been recently addressed by Dempsey, Liu, and Dempsey; Boersma and Dempsey; and by Gautschi. We show that, when $ \alpha $ is equal to a rational multiple of $ 2\pi $, these series can in the general case be summed in closed form in terms of known constants and special functions. General formulae giving $ {C_\nu }(\alpha )$ and $ {S_\nu }(\alpha )$ in terms of the generalized Riemann zeta function and the cosine and sine functions, respectively, are derived. Some simpler variants of these formulae are obtained, and various special results are established.

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Keywords: Summation of series, Riemann zeta function, generalized Riemann zeta function
Article copyright: © Copyright 1995 American Mathematical Society

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